Andrey Vasenko

School of Electronic Engineering,

Moscow Institute of Electronics and Mathematics,

Higher School of Economics

Majorana modes in 3D topological insulators with

warped surface state

Vyacheslav M. Silkin, DIPC, Spain

Eugeny V. Chulkov, DIPC, Spain

Collaborators

Alexander A. Golubov, Twente University, Netherlands

Bulk is insulating; edge (2D)/ surface (3D) a very good conductor.

Important ingredient: spin-orbit coupling:

opposite force for opposite spins.

Topological invariant is insensitive to any continuous deformation of

Hamiltonian (topological protection): disorder, geometry, weak interactions,

etc…

2D: HgTe/CdTe; 3D: Bi2Se3, Bi2Te3 Sb2Te3, TlBiSe2, Bi2Te2Se.

Theo1: C.L. Kane and E.J. Mele, PRL 95, 226801 (2005)

Theo2: B.A. Bernevig et al., Science 314, 1757 (2006)

Exp: M. König et al., Science 318, 766 (2007)

Theo: L. Fu, et al., PRL 98, 106803 (2007)

Exp1: Zhang H. et al., Nat. Phys. 5, 438 (2009)

Exp3: S. Takafumi et al., PRL 105, 136802 (2010)2D 3D

Examples:

…a new state of matter that has been predicted and discovered!

Topological insulator in a nutshell

S/TI proximity effect

J. Shen et al. (2013)

Symmetry classification of induced pair potential

Fermi-Dirac statistics

Symmetry of pair wave functions:

J. Linder et al., PRB (2008)

Majorana fermion

F. Wilczek, Nat. Phys. (2009)

Fu, Kane, PRL (2008)

Tanaka, Yokoyama, Nagaosa, PRL (2009)

Odd-frequency and Majorana

Asano, Tanaka, PRB (2013)

Hexagonal warping in 3D topological insulators

Fu, PRL (2009)

Mendle, Kotetes, Schon, PRB (2015)

Chen et al., Science (2009)

Li, Carbotte, PRB (2013)

Model: S/FI hybrid on TI surface

Bogoliubov – de Gennes – Dirac Hamiltonain

Green’s function (Nambu + spin space)

Anomalous Green’s function

singlet triplet triplet triplet

Bergeret, Volkov, Efetov, RMP (2005)

No warping

Anomalous Green’s function symmetry, Z is even in E and k

ESE

ETO

ETO

OTE

No warping

Anomalous Green’s function symmetry, Z is even in E and k

ESE

ETO

ETO

OTE

Finite warping

Anomalous Green’s function z-component

OTE

ETO

OTEETO

ESEOSO

Majorana fermion realization

Snelder, Golubov, Asano, Brinkman (2015)

Majorana fermion and warping

Vasenko, Golubov, Silkin, Chulkov, arXiv:1606.00905 (2016)

Spontaneous currents

Nesterov, Houzet, Meyer, arXiv:1512.03042 (2016)

Review

Thank you!

• We discuss singlet to triplet mixing in proximized 3D

topological insulators with warped surface state

• We establish the selection rule for Majorana Fermion

realization in S/FI structures formed on the surface of

the TI: S/FI boundary should be properly aligned with

respect to the snowflake contour.

• Spontaneous currents in S/TI hybrids.